Chaotic classical motion

It is clear, from the discussion thus far, that typical molecular Hamiltonians display features characteristic of chaotic classical motion. The logical order followed previously, that is, the introduction of well-defined concepts of chaotic behavior in classical ideal systems, followed by an examination of realistic molecular models, does not follow through to quantum mechanics. The primary difficulty is that quantum mechanics always predicts, for bound-state dynamics, quasiperiodic motion. Several aspects of quantum chaos are discussed in Section IV. We note at this point, however, that this quantum-... 

From a time-dependent point of view, recurrences in the probability of occupying the initially prepared state give rise to the fine structure in the overtone absorption spectrum. Though rudiments of these recurrences may be present in the short-time trajectory P n,t), chaotic classical motion destroys the longer time recurrences, which occur quantum mechanically. It is these latter recurrences which are needed to evaluate fine details in the absorption spectrum. Thus, the classical trajectory method may be limited to the evaluation of low-resolution absorption spectra. However, it should be pointed out that progress is being made in extracting information from systems with... 

The first classical trajectory study of iinimoleciilar decomposition and intramolecular motion for realistic anhannonic molecular Hamiltonians was perfonned by Bunker [12,13], Both intrinsic RRKM and non-RRKM dynamics was observed in these studies. Since this pioneering work, there have been numerous additional studies [9,k7,30,M,M, ai d from which two distinct types of intramolecular motion, chaotic and quasiperiodic [14], have been identified. Both are depicted in figure A3,12,7. Chaotic vibrational motion is not regular as predicted by tire nonnal-mode model and, instead, there is energy transfer between the modes. If all the modes of the molecule participate in the chaotic motion and energy flow is sufficiently rapid, an initial microcanonical ensemble is maintained as the molecule dissociates and RRKM behaviour is observed [9], For non-random excitation initial apparent non-RRKM behaviour is observed, but at longer times a microcanonical ensemble of states is fonned and the probability of decomposition becomes that of RRKM theory. 

G. Casati In the first part of your talk you discussed the longtime propagation of an initially localized quantum packet. Did you check whether in such situations the corresponding classical motion is in dynamically stable regions or instead is chaotic ... 

Indeed, the maximum time t up to which the quantum packet can propagate before its destruction depends on how far one is in semi-classical regions and scales in different ways depending on the nature of the corresponding classical motion. More precisely, we expect t (1 /h)a for dynamically stable systems and t In h for classically chaotic systems. 

In tetra-atomic molecules that are linear, the number of degrees of freedom is F =7, which again considerably complicates the analysis. Nevertheless, the dynamics of such systems can turn out to be tractable if the anharmonicities may be considered as perturbations with respect to the harmonic zero-order Hamiltonian. In such cases, the regular classical motions remain dominant in phase space as compared with the chaotic zones, and the edge periodic orbits of subsystems again form a skeleton for the bulk periodic orbits. 

The analysis of quantum mechanical eigenfunctions in energy regimes where the classical motion is chaotic (Heller 1984, 1986 Founargio-takis, Farantos, Contopoulos, and Polymilis 1989). 

The fundamental assumption of RRKM theory is that the classical motion of the reactant is sufficiently chaotic so that a micro-canonical ensemble of states is maintained as the reactant decomposes [6,324]. This assumption is often referred to as one of a rapid intramolecular vibrational energy redistribution (IVR) [12]. By making this assumption, at any time k E) is given by Eq. (62). As a result of the fixed time-independent rate constant k(E), N(t) decays exponentially, i.e.. 

In this section we will briefly consider a few questions concerning the expected accuracy of trajectory calculations in the context of unimolecular dissociation processes. For a particular molecular system it depends on the size of the system (number of atoms), the energy regime, the nature of the classical motion (e.g. regular or chaotic) and, very importantly, the timescale. These issues will be addressed in the following discussions of both random and non-random excitation of molecules. 

Exponential divergence in systems that are chaotic prevents accurate long-time trajectory calculations of their dynamics. That is, numerical errors18 propagate exponentially during the dynamics so that accuracy beyond 100 characteristic periods of motion is extraordinarily difficult to achieve thus, accurate long-time dynamics is essentially uncomputable for chaotic classical systems. This serves as additional... 

The phase space of a coupled, two-identical-anharmonic oscillator system is four-dimensional. Conservation of energy and polyad number reduces the number of independent variables from four to two. At specified values of E and N = vr + vl = vs+ v0 (in classical mechanics, N need no longer be restricted to integer values nor E to eigenenergies), accessible phase space divides into several distinct regions of regular, qualitatively describable motions and (for more general dynamical systems) chaotic, indescribable motions. Systematic variation of E and N reveals bifurcations in the number of forms of these describable motions. Examination of the classical mechanical form of the polyad Heff often reveals the locations and causes of such bifurcations. 

It is also interesting to consider the classical/quantal correspondence in the number of energized molecules versus time N(/, E), Eq. (8.22), for a microcanonical ensemble of chaotic trajectories. Because of the above zero-point energy effect and the improper treatment of resonances by chaotic classical trajectories, the classical and quantal I l( , t) are not expected to agree. For example, if the classical motion is sufficiently chaotic so that a microcanonical ensemble is maintained during the decomposition process, the classical N(/, E) will be exponential with a rate constant equal to the classical (not quantal) RRKM value. However, the quantal decay is expected to be statistical state specific, where the random 4i s give rise to statistical fluctuations in the k and a nonexponential N(r, E). This distinction between classical and quantum mechanics for Hamiltonians, with classical f (/, E) which agree with classical RRKM theory, is expected to be evident for numerous systems. 

Gaspard and Rice have studied the classical, semiclassical and full quantum mechanical dynamics of the scattering of a point particle from three hard discs fixed in a plane (see Fig. 11). We note that the classical motion (which is chaotic) consists of trajectories which are trapped between the discs. 

The fundamental assumption of RRKM theory is that the classical motion of the reactant is sufficiently chaotic to maintain the microcanonical ensemble of... 

In classical Newtonian physics the elementar volume of a configurational space cell is infinitesimal (it looks like Planck s constant ti is accepted to be zero) the electron distribution upon their energy is given by Maxwell-Boltzmann statistics there are large amounts of particles, all of which tend to occupy the state with the lowest energy, though chaotic temperature motion, on the other hand, scatters them on different energies. This process is described by the Boltzmann factor. 

Shirts R B and Reinhardt W P 1982 Approximate constants of motion for classically chaotic vibrational dynamics vague tori, semiclassical quantization, and classical intramolecular energy flow J. Cham. Phys. 77 5204-17... 

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